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If we observe two galaxies from the Earth that are diametrically opposed and each 10,000 light-years from the Earth, will the separation distance between the galaxies be 20,000 light-years?

Really, the question is: if the galaxies instead were separated from Earth by 10 billion light-years, would the separation distance between the galaxies then be 20 billion light-years?

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I would assume so. I didn't see the difference between the first and second question except the time difference. Please clarify –  Jus12 Aug 31 '11 at 17:33
    
The first case is far too close, galactic dimensions are typically of the order of 100,00ly or more. Get within that range and we are dealing with mergers and collisions. For great distances distance does't have the usual meaing, since the spacetime the galaxies are embedded in is expanding. –  Omega Centauri Aug 31 '11 at 18:44
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This is much more complicated than it seems. Keep in mind, the Universe keeps expanding, so even as one photon leaves the galaxy on its way towards us, that galaxy has already moved a bit further away.

So, as you measure the red shift and whatnot, all you could say is "these particular photons have traveled 10 Gly to reach us", but you can't say "the distance right now from here to that galaxy is 10 Gly". That galaxy is already further away by now.

To complicate things even more, this is a relativistic universe, so the word "now" does not have a unique, generally-agreed-upon meaning for all observers. Your "now" is different from my "now", which is different from the "now" of someone living in the Andromeda galaxy. This is not because we don't measure time precisely enough, but it's an inherent attribute of time in a relativistic universe. Relativistic space-time is not this crystal clear background we're moving through, but it's more like molasses or a rubber sheet, it flows and bends depending on what you do or where you are.

To answer your question specifically, all you could say is "the image of this galaxy, from my perspective 'now', is at 10 Gly; the image of that other galaxy, from my perspective 'now', is at 10 Gly in the opposite direction; so these images, seen from my particular point in space and time, are as if they are separated by 20 Gly".

You could calculate where those galaxies are "now", from your perspective, and you may even deduce that they are at 12 Gly each, so therefore they are separated "now" by 24 Gly. But, again, another observer, in a different place in space-time or phase-space, may obtain a different result - and both results, yours and theirs, are equally correct. The difference will likely be very small, unless extremely high velocities or energies are involved, but be aware that it exists.

EDIT: I assumed a more or less flat space-time in all of the above. If that assumption is not correct, then that's another source of fuzziness for the results.

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wow Florin.. excelent answer. thank you very much!!! –  jormansandoval Aug 31 '11 at 21:27
    
I think the another distance to consider is the redshift that inhabitants of one galaxy will measure of the other. i.e. if we see galaxies A and B diametrically opposed at redshifts corresponding to 10 Gly, what redshift distance would galaxy A measure of galaxy B? Much more than 20 Gly. –  Warrick Sep 1 '11 at 17:09
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